Integrated Frequency – Part 2
In Part 1 of Integrated Frequency, I used frequencies derived from the R20 (root 20Hz) family, with the fundamental below human hearing, depicting rhythm (64 octaves down) as a whole-note. This frequency tree very closely approximates the key of E Major, in Equal Temperament.
In order to modulate, from a traditional sense, to the other eleven keys around the Circle of Fifths, I initially took the R20 frequency and multiplied its 3rd node (that is, the fifth scale degree). In other words, 20Hz x 3 = 60. 60Hz, then would be the next key center (B major). By doing this, sequentially around the circle of fifths, the frequency becomes fractionally deviated from the original 20Hz. (The Pythagorean Scale is derived this way.)
E 20 x 3 = 60/2 = 30
B 30 x 3 = 90/4 = 22.5
F# 22.5 x 3 = 67.5/2 = 33.75
C#/Db 33.75 x 3 = 101.25/4 = 25.3125
Ab 25.3125 x 3 = 75.9375/2 = 37.96875
Eb 37.96875 x 3 = 113.90625/4 = 28.4765625
Bb 28.4765625 x 3 = 85.4296875/2 = 21.357421875
F 21.357421875 x 3 = 64.072265625/2 = 32.0361328125
C 32.0361328125 x 3 = 96.1083984375/4 = 24.027099605
G 24.027099605 x 3 = 72.0812988282/2 = 36.0406494141
D 36.0406494141 x 3 = 108.1219482423/4 = 27.030487060
A 27.030487060 x 3 = 81.0914611818/4 = 20.2728652955
Each time I multiplied the fundamental, I would reduce it by octaves down to as close to 20Hz (the threshold of human hearing) as possible, so as to show the relationships between the key centers within an octave range. Octave shifts are accomplished simply by multiplying or dividing by 2.
The difference from the starting frequency of 20Hz through the entire cycle back to the 20Hz region is .2728652955. I decided at this point to round the numbers to their nearest whole number. I had some logistical reasons for doing so. For one, to work with the above frequencies, dropping the octaves down below human hearing, in order to create congruent tempo families, would be unmanageable. Secondly, Ableton currently only allows tempo adjustments to the second decimal point.
The rounded numbers were as follows:
E 20
B 30
F# 23
C#/Db 34
Ab 26
Eb 38
Bb 29
F 21
C 32
G 24
D 36
A 27
However, doing this was a compromise to my original desire to integrate tempo and pitch with precision.
I then endeavored to derive a 12-note chromatic scale from within the original 40Hz overtone scale, finding pitches matching the chromatic scale as close to the fundamental as possible.
The nodes in chromatic order are as follows:
Chr. # Key Hz Node #
1 E 40 1
2 F 680 17
3 F# 360 9
4 G 760 19
5 G# 200 5
6 A 880 22
7 A# 920 23
8 B 120 3
9 C 1040 26
10 C# 1080 27
11 D 280 7
12 D# 600 15
Reducing these frequencies down to the THH (threshold of human hearing) by octaves, produced the following:
E 20Hz
F 21.5
F# 22.5
G 23.75
G# 25
A 27.5
A# 28.75
B 30
C 32.5
C# 33.75
D 35
D# 37.5
E 40
I discovered that this sequence had a palindromic shape to its intervallic structure:
It is interesting to note that the differences between the Equal Temperament tuning, Outside the R20/40 Overtone Scale, and Inside the R20/R40 Overtone Scale are miniscule.
As shown in the first Integrated Frequency article, each pitch-center has its corresponding tempo. And each tempo can have allocated rhythmic values. Following is the Tempo Frequency Scale for 18.75BPM (Remember that this number is 64 octaves below 20Hz):
Each Chromatic note comprising the Circle of Fifths, derived from within the original Overtone Scale of 20/40Hz can have its own corresponding Rhythmic Frequency Scale, such as the one shown above.
The quarter-note tempo of each Chromatic scale degree is as follows:
E 75 (BPM)
F 80.625
F# 84.375
G 89.0625
G#/Ab 93.75
A 103.125
Bb 107.8125
B 112.5
C 121.875
C#/Db 126.5625
D 131.25
D#/Eb 140.625
E 150
(It is interesting to note that these tempos are 16 octaves below the notes which live right at the THH line.)
From this, a Chromatic Tempo Scale can be constructed:
My goal was to find tempos usable in Ableton Live, which necessitates numbers only with two decimal points. The rhythmic designation (i.e. quarter-note, half-note, etc.) of each of these is arbitrary. The main idea is that, whatever the numerical ‘octave’ they fall into, that they are all in line with the same tempo family. (I.e. 37.5 = 75 =150)
Hence, the usable tempos (for the sake of Ableton) are as follows:
E 20Hz quarter = 75/150BPM
A 27.5 eighth = 206.25
D 35 quarter = 131.25
G 23.75 sixteenth = 356.25
C 32.5 sixteenth = 243.75
F 21.5 quarter = 161.25
Bb 28.75 sixteenth = 431.25
Eb 37.5 sixteenth = 281.25
Ab 25 quarter = 93.75
C#/Db 33.75 sixteenth = 506.25
F# 22.5 eighth = 168.75
B 30 quarter = 56.25/112.5
I endeavored to create within one synthesizer the possibility to transition between the various octaves (3 with the fundamental above human hearing, and 2 below human hearing) in all 12 keys. In order to do so, I created the following chart. Each of the frequencies below represents a 32-node Overtone Scale. The following example is of the overtone series within the key of E, starting on 20Hz. The information is in a coll file, used in Max. (The MIDI note number is on the left, and frequency on the right.)
Every key has five overtone scales (as the example above), each an octave apart. There are twelve keys, all derived from within the original overtone series of 20Hz. Each key has its own tempo, as discussed above. This produces a total of 60 scales.
(The corresponding tempos for each scale are usable within Ableton’s tempo limitations.)
In addition, I added a multi-octave chromatic scale for each of the 12 key centers, using the same frequency ratios derived from within the original R20 Overtone Scale.
(In the following chart, octaves are horizontal, and keys are vertical, starting with the key of E at 20Hz):
All scales used totaled 72. Using Max/MSP, I created each of these scales to be quickly accessible from within the synthesizer.
The result of this gives access, not only to all chromatic frequencies used in Western composition, but also their corresponding microtonal frequencies via the corresponding Overtone Scales. In addition to this, all frequencies used are derived from the same original R20 Overtone Scale, making them all universally harmonically congruent. This is not Equal Temperament, but rather a system which includes Equal Temperament’s chromatic functionality, yet also with the possibility of comprehensive harmonic unity of the Overtone System.
It is important to note that this methodology could be used, based upon any chosen frequency. The frequency I chose was based upon its nominal simplicity, and also due to a feature important to me: to include the ‘anchor’ frequency of A440. When dealing with a subject as vast as all possible frequencies, I found it necessary to have an anchor-point within the system. The system itself could be expanded to include any number of different unrelated or related base-frequencies. The point is not which base-frequency is used, but rather a comprehensive system of integration, harmonically and rhythmically.
It is also interesting to note that it could be of substantial importance for any musical genre to be aware of the tempo/pitch relationship for the key they are performing within, adding greater continuity to the structure of their performance or production. Correlating key and tempo could be a new breakthrough in performance technique, as well as in the production/engineering of any musical style.
Also, if the software tools become available to do so, one could combine a system like the one discussed above with added expansion to the Pythagorean-derived circle of fifths. In this way, there would be 3 tiers of harmony, each nested inside the larger organizational sphere, as follows: 1) Harmony derived within a single overtone scale, 2) 12 overtone scales, via the circle of fifths, derived from within the overtone scale (as enumerated above), and 3) 12 sets of overtone scales, via the circle of fifths, derived from outside the overtone scale, using the Pythagorean system.
The example I created for myself, described above, uses 72 possible overtone scales, all derived from R20 (which is basically the key of E). That set of scales could be expanded, using the Pythagorean approach, deriving a circle of fifths outside the overtone scale, essentially giving a palette of scales (i.e. 72 x 12). In other words, within each key center, one could have 72 possible tunings, all inter-related.
I have not created a system in this way, since the technology in current DAW software does not yet provide this kind of accuracy in pitch to tempo relationships. But the possibility exists, nonetheless.
Following are experiments using the above system.
Harmonic Modulation
In this experiment, I took harmonic progression material from an earlier project, derived from within the single overtone scale of E.
Modulating through all 12 chromatic keys, and utilizing rhythmic sounds from fundamental frequencies of R20 (root, 20Hz), the thematic harmonic progression is stated three times in three different tempi, all inter-related.
2 - Key Counterpoint
This is an experiment using two distinctly composed lines, each in its own key. The keys are R70 (D Major) in the tempo of 131.25BPM and R40 (E Major) in the tempo of 150BPM. The lines interact in an 8-bar pattern in a 7:8 relationship.
Interestingly, even though the lines are similar in timbre, the tuning of each line causes the ear to psycho-acoustically ‘track’ with the lines without confusing them. In panning, they end up trading places, and the ear can follow the motion.
Full-range Melody, 3 Keys
In this experiment, a melodic phrase is used in three different keys: F#, B, and E. The corresponding tempi relating to each key is as follows:
F# = 168.75BPM
B = 112.5BPM
E = 150BPM
The tempo/scale relationships formed between each of the parts create a macro-polyrhythm of 9:12:16 as the melodies repeat. Each of the relationships constantly change as they inter-weave.
Bi-tonal Melody with Harmony
This is an experiment in modulation between the key/tempos of R40 (eighth-note = 150) and R27.5 (eighth-note = 206.25). The melody, harmony, and tempi shift from one key to the other, alternating back and forth, from zone to zone. Creating tuplets in the melody, also gives rhythmic dimensionality to the changing tempi. (As stated earlier, rhythm also is congruent with the overtone series, symbolically portraying 1, 2, 3, etc. as quarter, eighth, triplet eighth, etc.)
Modulation with Serial (P, R, I, R/I)
This is an experiment in modulation between the three overtone series ‘keys’ of R30 (B), R20 (E), and R22.5 (F#). The chord progression sequences happen within a tonic, dominant, and subdominant framework.
In addition, I added a 12-note chromatic line, developed with the common serialism components of Prime, Retrograde, Inversion, and Retrograde/Inversion.
As the keys shift, the tempi and 12-tone line all shift, as well. Since the line is derived directly from the same overtone series of its underlying harmonies, we see that serialism and harmony can co-exist.
In this work, the elements of tempo, harmony, and chromatic serialism are compatible and congruent.
LH 2022-03-04